Multicomponent Distributed Acoustic SensingIvan Lim Chen Ning & Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

Distributed Acoustic Sensing (DAS) data in boreholes are notcommonly used for reservoir characterization because multi-component data are needed to quantify different wave modes.We propose two approaches to obtain multicomponent DASdata by using either multiple parallel optical fibers or a heli-cal optical fiber. The multiple parallel optical fibers use theshape sensing method which is based on measurements of ax-ial strain gradient to obtain the curvature of the cable and thenreconstruct the components of the displacement field. How-ever, simply using curvature measurements does not providesufficient information to reconstruct the displacements for dif-ferent wave modes. The helical optical fiber uses the char-acteristics of the helix shape to obtain angle dependent strainmeasurements, which provides sufficient information to recon-struct the strain tensor of the surrounding field.

INTRODUCTION

Distributed Acoustic Sensing (DAS) is rapidly gaining popu-larity in the oil and gas industry especially for Vertical SeismicProfile (VSP) imaging and for reservoir monitoring (Mestayeret al., 2011; Cox et al., 2012; Mateeva et al., 2012, 2013; Daleyet al., 2013; Madsen et al., 2013). The advantages of DAS forborehole applications in terms of costs, the deployment mecha-nism, and spatial resolution make its usage attractive over con-ventional geophone acquisition (Lumens et al., 2013; Mateevaet al., 2013).

Lumens (2014) indicates that the DAS system is more sensitivein the axial direction compared to the radial direction, thus re-ducing the DAS measurable data to the axial strain which canbe acquired with acceptable signal-to-noise ratio. Publishedwork does not detail mechanisms for extracting multicompo-nent data from the axial strain measurement using DAS. Toacquire multicomponent data, alternative measuring devicessuch as geophones are deployed, which are costly and do notprovide the dense spatial sampling of DAS. In this paper, weinvestigate possibilities to use different optical fiber configura-tions to obtain multicomponent information from DAS data.

Our first approach is to use optical fiber as a shape sensingtool. The relationship between bending and the difference instrain between two Fiber Bragg Gratings (FBG) has been dis-cussed by Gander et al. (2000) as a curvature sensor. Flockhartet al. (2003) and Fender et al. (2006) demonstrate positive ex-perimental results for using this theory as a curvature sensorfor structural monitoring. However, their discussion is limitedto using the optical fiber as a shape monitoring tool. We ex-ploit shape sensing capabilities to obtain data components thatcannot be measured using the axial strain along a single fiber.

Our second approach is to use a helical-shaped optical fiber.

Using the characteristics of the helix and axial strain measure-ment of the optical fiber, we can calculate the entire strain ten-sor at every measuring location under the assumption that theseismic wavelength is much longer than the helix period. Weshow the theoretical relationships between the measured axialstrain in the optical fiber and the full strain tensor in the sur-rounding area, and demonstrate the applicability of this strat-egy using a synthetic example.

MULTIPLE PARALLEL OPTICAL FIBERS

The axial strain measurement by the optical fiber is a projec-tion of the strain tensor from the surrounding area as a functionof the position of the optical fiber. Using the optical fiber sys-tem illustrated in Figure 1a, we can measure the bending oftwo optical fiber by the relationship between their differencein axial strain and curvature κ (Gander et al., 2000)

κ =∆εmm

r, (1)

where ∆εmm is the difference in strain and r is the distance be-tween two optical fibers. At least two optical fibers are neededto resolve bend measurement of one axis. We show the n-axisbending calculation for two horizontal optical fibers denotedwith the superscript top and bottom

κ =εbottom

mm − εtopmm

2∆n. (2)

We recognize that equation 2 effectively represents the cen-tered finite-difference approximation of the transverse deriva-tive of the axial strain

κ =εmm (n+∆n)− εmm (n−∆n)

2∆n≈ ∂εmm

∂n. (3)

l

n

m

(a)

l

n

reiθkcos(θ)θ

ksin(θ)ϕ

κ1κ2

κ3

κ

(b)

Figure 1: (a) An example of 3D parallel fiber optic configura-tion with local coordinates and (b) the cross section of a cablewith (b) three optical fiber core (circles) with respective cur-vature vectors (solid line). The resultant vector (dotted line) isthe sum of the respective curvature vectors.

We can generalize this idea to 3D by calculating the axial straingradient between the optical fibers. Moore and Rogge (2012)

Multicomponent Distributed Acoustic Sensing

show that the 2-axis bend measurement calculation can be ob-tained with an arbitrary circular arrangement. An example ofthe arrangement is shown in Figure 1a. The expression neededto calculate local curvature vector for the respective opticalfibers is (Moore and Rogge, 2012)

κκκ i =ε i

mm2∆n

(cos(θi) n̂+ sin(θi) l̂

), (4)

where i is the optical fiber index, θi is the angle between thepositive l-axis of the respective optical fiber as shown in Fig-ure 1b. Expression equation 4 decomposes the scalar curvatureinto a vector curvature based on a local coordinate system ofthe cable as the reference axis. The resultant curvature vectorκκκ represents the overall bending of the optical fiber by sum-ming all the curvature vectors of the respective optical fibers

κκκ =

N∑i=1

κκκ i . (5)

Calculating the angle made by κκκ with respect to the referenceaxis gives

ϕϕϕ = tan−1(

κn

κl

), (6)

and its derivative of the angle we can obtain the torsion of theoptical fibers as

τ =dϕ

dm. (7)

The curvature κκκ allow us to compute the displacement compo-nents through the geometrical definition

κκκ =∂ 2ul

∂m2 l̂ +∂ 2un

∂m2 n̂ . (8)

Solving equation 8 using the calculated curvature yields thetransverse displacement components (ul and un). The axialdisplacement um is the integration of the measured axial strainεmm along the optical fiber. Therefore, we can obtain all thedisplacement components with respect to the local coordinatesystem of the optical fiber given that we have at least threeparallel optical fibers.

Plane wave exampleWe consider planar P and S waves impinging at an angle onthe optical fiber to illustrate the measures of axial strain. Thisexample shows compression and elongation along the opticalfiber for both wave modes which is observed through the ax-ial strain tensor shown in Figures 2a and 2b. The axial strainfor both P and S waves show that they share the same signalthough they are different in terms of polarization, thereforeboth P and S waves share the same gradient. As the curvaturemeasurement is ultimately the gradient of axial strain in thetransverse direction, P and S waves have the same curvatureto calculate the displacement un in the n-direction as shownin Figures 3c and 3d. The consequence of the similar gradi-ent of axial strain, we are unable to reconstruct the transversedisplacement component with the correct sign for S-waves asillustrated by Figure 3f where the S-wave points in the direc-tion of P-waves.

(a) (b)

Figure 2: Strain tensor for plane (a) P-wave and (b) S-wavepropagating with an incidence angle of 30◦.

DiscussionUsing the analytical plane waves example, we show the lack ofinformation using the axial strain itself when we calculate thecurvature for both P and S waves that are used to reconstructthe displacement components. The axial strain does not con-tain information to evaluate the polarization direction of differ-ent wave modes. This creates an ambiguity in reconstructingthe displacement field.

(a) (b)

(c) (d)

(e) (f)

Figure 3: The control (a) P-wave (arrow pointing top right) and(b) S-wave exaggerated displacement vectors (arrow pointingbottom right). The scaled curvature (dotted line) and displace-ment (solid line) for (c) P-wave and (d) S-wave. The recon-structed exaggerated displacement vectors for (e) P- and (f)S-waves using the calculated curvature.

Our results show the calculated curvatures for P and S waveshave the same sign as a result of the same sign for the measuredaxial strain. The reconstructed displacement shares the samesign as the curvature. However, the analytical displacementresults for P- and S-wave do not share the same sign. The

Multicomponent Distributed Acoustic Sensing

disagreement between the curvature sign and the displacementfor S-wave is not a result of an incorrect curvature calculation,but that of insufficient information to provide the reconstructthe displacement with the appropriate sign.

Based on the reconstructed displacement results, multiple par-allel optical fibers do not give us enough information to accu-rately reconstruct the multicomponent DAS data. In the fol-lowing section, we propose a different optical fiber layout thatcan provide adequate information, under certain assumptions,for multicomponent DAS data reconstruction.

HELICAL OPTICAL FIBER

A helical optical fiber configuration for Distributed AcousticSensing (DAS) (Den et al., 2013) is designed to detect broad-side acoustic signals, which refers to waves that arrive perpen-dicular to the direction along the optical fiber. Here, we exploitthe helical shape as a tool to measure different projections ofthe strain field along the optical fiber.

windingdirection

n

m

l

z

y

x

Figure 4: An example of the helical optical fiber with the as-sociated local coordinate system.

In this configuration, we make the assumption that the wave-length of the wavefield is much greater than the spatial wave-length of the helical optical fiber. This assumption is importantso that we can group multiple measurements along the opticalfiber to refer to the same strain tensor field.

The relationship between the axial strain measured by the op-tical fiber and the strain tensor of the surrounding area via theangles can be expressed as (Young and Budynas, 2002)

εεε′ = RεεεRT , (9)

where the symbol (′) denotes rotated strain tensor and R repre-sents the rotation matrix. Rotation of a coordinate system canbe written

x′m1 l1 n1m2 l2 n2m3 l3 n3

=

RR11 R12 R13R21 R22 R23R31 R32 R33

x1 0 0

0 1 00 0 1

, (10)

where x′ denotes axes of the rotated coordinate system and xrepresents axes of the original coordinate system which is anidentity matrix. Therefore, the rotation matrix is representedby the local coordinate system of the optical fiber. Since theprimary strain measurement of the optical fiber is due to axialstrain only, we can narrow the problem to the rotated axial

strain measurement in 3D at a location along the optical fiberas follows

d[εmm]=

G[R2

11 R212 R2

13 2R11R12 2R11R13 2R12R13] m

εxxεyyεzzεxyεxzεyz

, (11)

where the observed data d vector represents axial strain mea-surements along a segment of the optical fiber of length hwhich is assumed to be much smaller than the wavelength ofthe wavefield. The matrix G represents the expanded rotationmatrix from equation 10. The number of rows in the data vec-tor and matrix G refers to the number of measurements thatrefers to the same strain tensor of the surrounding area.

The expression in equation 11 can be solved using least-squaresinversion where the kernel or forward operator (G) is known.The data (d) are the recorded axial strains along a segment ofthe helical optical fiber (this gives axial strain as a functionof angles, given the assumption that the seismic wavelength ismuch larger than the helix period holds) and the model (m) isthe strain tensor of the surrounding area. Therefore, the modelcan be expressed as

m =(

GTG)−1

GTd . (12)

The number of axial strain measurements with the associatedangles can be varied accordingly with the parameters of thehelix such as the radius of the helix, the distance between thewindings and the interval at which the measurements are made.In Figure 5f, we show an example of the number of measur-able axial strain with the respectively associated angles. Theexample is designed with a radius and a distance between thewindings which are much smaller than the wavelength of thewavefield. Using the same parameters, Figure 5e shows a 3Dexample with the associated angle and azimuth of the mea-sured locations along the fiber. The 3D plot shows a constantmeasuring angle due to the characteristic of the helix with aconstant wrapping angle thus, this configuration could not pro-vide a range of angle dependent measurement to determine thestrain tensor of the surrounding area.

By introducing a varying winding distance along the fiber, weobtain a better coverage for the measured angle dependent ax-ial strain. This alternative configuration in 2D represents achirp function as shown in Figure 5d. Using the chirping helixmeasurements, we can solve equation 12 to obtain the straintensor of the surrounding area.

Plane wave exampleAssuming that the seismic wavelength is much larger than thehelix period, we are able to measure axial strain εmm(θ) as afunction of angle. The strain tensors of P- and S-waves are dif-ferent as shown in Figure 7a and 7b. Therefore, the measuredaxial strain as a function of angles for P- and S-waves are dif-ferent due to the contribution of all the elements in the straintensor as expressed in equation 11. An example of a rotatedstrain field at a specific time for all angles is shown in Fig-ure 6a for P-wave and Figure 6b for S-wave that illustrates the

Multicomponent Distributed Acoustic Sensing

zx

y

(a)

z

x

(b)

z

x

y

(c)

z

x

(d)

(e)

0◦

315◦

45◦

270◦

90◦

135◦

225◦

180◦

(f) (g)

0◦

315◦

45◦

270◦

90◦

135◦

225◦

180◦

(h)

Figure 5: Example of a helical configuration for a constantwrap angle in (a) 3D and (b) 2D together with the (e) anglesand azimuth for 3D and (f) angles for 2D. An example of achirping helical configuration in (c) 3D and (d) 2D togetherwith the (g) angles and azimuth for 3D and (h) angles for 2D.

difference between the wave modes. By using the observed ro-tated axial strain data and solving equation 12, we successfullyreconstruct the strain field for the plane P-wave (Figure 7c) andthe S-wave (Figure 7d).

(a) (b)

Figure 6: Axial strain of optical fiber for plane (a) P-wave and(b) S-wave with a wavelength of 100m as a function of anglesθ at a specific time. The difference in the rotated strain fieldof P- and S-wave is due to the difference its respective straintensor contributing in the projection to the axial strain of theoptical fiber.

DiscussionOur synthetic example indicates that we can reconstruct thestrain tensor field if we have the angle dependent axial strainmeasurements. As seen in Figure 5f and 5h, the chirping helixcan provide a wider range of angles with respect to the wind-ing direction of the traditional helix with a constant wrap an-gle. In situations where the period of the helix is equivalentto the Distributed Acoustic Sensing (DAS) system measuringdistance ∆h, the traditional helix can only provide a constantangle for the axial strain measurement whereas the chirpinghelix can provide a range of angles due to the change in thehelix coil frequency.

(a) (b)

(c) (d)

Figure 7: Strain field for plane (a) P-wave and (b) S-wavepropagating at 30◦. The reconstructed (c) P-wave and (d) S-wave by solving the least-squares inversion problem using theangle dependent axial strain measurements. The vertical axisis time and the horizontal axis is the receivers.

CONCLUSIONS

Multiple parallel optical fibers or a single helical optical fiberhave the potential to provide multicomponent DAS data. How-ever, the shape sensing method applied by the parallel opticalfibers is unable to accurately reconstruct correct displacementswhen multiple wave modes are involved. The reliance of thismethod on the gradient of the axial strain with respect to mul-tiple parallel optical fibers is insufficient due to lack of addi-tional information to relate curvature to the polarization of theincident wavefields.

However, the strain tensor can be reconstructed given enoughangle dependent axial strain measurement associated with a fixposition along the optical fiber. This is true even for multiplewave modes as the strain tensor of the respective wave modescontributes to the axial strain along the optical fiber. The an-gle dependent measurement can be obtained using the helicaloptical fiber under the assumption that the seismic wavelengthis much larger than the helix period. Alternative to the opticalfiber helix shape allow us to measure a wider range of angledependent axial strains compared to the traditional helix andcan reduce the possibility of repeated angle dependent mea-surements.

ACKNOWLEDGMENTS

We thank the sponsors of the Center for Wave Phenomena,whose support made this research possible. The reproduciblenumeric examples in this paper use the Madagascar open-sourcesoftware package (Fomel et al., 2013) freely available fromhttp://www.ahay.org.

Multicomponent Distributed Acoustic Sensing

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